Math 14 Lecture Notes Ch. 6.1

Transcription

1 6.1 Normal Distribution What is normal? a 10-year old boy that is 4' tall? 5' tall? 6' tall? a 25-year old woman with a shoe size of 5? 7? 9? an adult alligator that weighs 200 pounds? 500 pounds? 800 pounds? rolling a sum that is odd using 2 dice 200 times in 300 rolls? Many variables follow what we call a normal distribution. The eamples above are normal in distribution and thus the questions of normality can be answered. g A normal distribution is a distribution of a continuous random variable and is symmetrical with the mean, median and mode coincident. We denote a normal distribution as X ~ N(µ, σ). g A normal curve is the graph of a normal distribution and has the following properties: bell-shaped symmetric about a vertical line through its center continuous etends infinitely in both directions, always getting closer to the -ais, but never touching it the total area under the normal curve is 1 the area under the curve that lies within 1 standard deviation is approimately.68, within 2 standard deviations is approimately.95, and within 3 standard deviations is approimately.997 (known as the Empirical Rule) ( X µ ) 2 2σ 2 follows the equation y = e σ 2π where µ = population mean and σ = population standard deviation µ 3σ µ 2σ µ σ µ µ + σ µ + 2σ µ + 3σ Page 1 of 10

2 Eample 1: Intelligence Quotient Scores The probability distribution of IQ scores is a very close approimation to a normal distribution. Construct a probability histogram for the data given Range Percent % % % % % % % % The graph to the right illustrates the normal appearance of a large number of IQ scores. The mean IQ score is 100 and standard deviation is 15, hence, the IQ distribution is denoted X ~ N(100, 15). Page 2 of 10

3 As the class width decreases, the bell shape of normally distributed data becomes more smooth. Class width: 4 Class width: 2 Class width: 1.3 Class width: 1 Class width: 0.8 Class width: 0.67 Class width: 0.57 Class width: 0.5 Class width: 0.44 Page 3 of 10

4 Eample 2: In this "living" histogram of 175 cadets from the Connecticut Agricultural College, ROTC, we can see an approimation to the normal curve. Below is a histogram constructed from this model What is the probability that a randomly selected cadet from this group of 175 will have a height greater than 70 inches? P (X > 70) = 2. What percent of the group is shorter than 68 inches? Percent shorter than 68 = 3. What are the heights of the cadets in the shortest 1% of this group? Number of cadets in shortest 1% = = Heights of shortest 1% are 4. What are the heights of the cadets in the tallest 5% of this group? Number of cadets in tallest 5% = = Heights of tallest 5% are Page 4 of 10

5 A normal curve can vary in shape according to the variation of its mean and standard deviation. Compare the 4 normal curves below. The curve with the smallest standard deviation has the tallest peak and narrowest tails. The curve with the largest standard deviation has the shortest peak and widest tails. The curve with the smallest mean ( 2) lies to the left of the others. The blue, red, and yellow curves all have a mean of 0 Compare the graphs at right with the data in the table below: Heights of American Adults (in inches) Women µ = 63.8 σ = 2.5 Men µ = 69.3 σ = 2.8 Page 5 of 10

6 What does not vary is the empirical rule for all normal distributions: 68% of the data lies within 1 standard deviation of the mean 95% of the data lies within 2 standard deviations of the mean 99.7% of the data lies within 3 standard deviations of the mean It is helpful to standardize all normal curves in order to analyze certain attributes of a given normally distributed data set. We can perform a translation on any given normal curve that will set its mean at 0 and its standard deviation at 1 by using the formula z = X µ σ. The standard normal distribution is a normal distribution with mean, 0, and standard deviation, 1. Compare the difference between a normal curve and standard normal curve. We call these z-scores. Page 6 of 10

7 To analyze the data of a normal distribution, we often need to find the area between the curve and the horizontal ais. Recall that although the curve never touches the -ais and it etends infinitely in both directions, the area beneath it is 1. Consider the following illustration that = 1 1. Divide the square vertically into two pieces and label the left side area as. 2. Divide the right half horizontally into two pieces and label the bottom part area as. 3. Continue this division and labeling process a few times. Notice that this process can continue indefinitely without eceeding the square, yet always adds half the previously added area. Thus, = 1. 8 Here is another illustration of the series Use the following grid to draw and shade rectangles with areas of 1 2, then 1 4, then 1 8, etcetera. The first one has been drawn for you The normal curve does not follow this same progression, but one can see how a finite area can be stretched to infinity from this activity. Page 7 of 10

8 Now, we will eplore the area under the normal curve using the symmetry of the normal curve and the Empirical Rule. 5. Using the empirical rule, shade approimately 68% under the curve. 6. Using the empirical rule, shade approimately 34% under the curve. 7. Using the empirical rule, shade approimately 95% under the curve. 8. Using the empirical rule, shade approimately 5% under the curve. 9. Using the empirical rule, shade approimately 84% under the curve. 10. Using the empirical rule, shade approimately 2.5% under the curve. 11. Using the empirical rule, approimately what percent under the curve has been shaded? 12. Using the empirical rule, approimately what percent under the curve has been shaded? 13. Using the empirical rule, approimately what percent under the curve has been shaded? 14. Using the empirical rule, approimately what percent under the curve has been shaded? Page 8 of 10

9 Use the z-score table at the end of this document to answer the following: 15. Find the percent of the area that lies under the curve and between z = 0 and z = Find the percent of the area that lies under the curve and between z = 1.33 and z = Find P(0 < z < 2.32) 18. Find P(z > 2.32) Page 9 of 10

10 z Page 10 of 10